A family of functional inequalities

For displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Lojasiewicz inequalities. In a second part, we specialise these inequalities to some classical geodesically convex functionals. For the Boltzmann entropy, we obtain the equivalence between logarithmic Sobolev and Talagrand's inequalities. On the other hand, the non-linear entropy and the Gagliardo-Nirenberg inequality provide a Talagrand inequality which seems to be a new equivalence. Our method allows also to recover some results on the asymptotic behaviour of the associated gradient flows

A family of functional inequalities

For displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Lojasiewicz inequalities. In a second part, we specialise these inequalities to some classical geodesically convex functionals. For the Boltzmann entropy, we obtain the equivalence between logarithmic Sobolev and Talagrand's inequalities. On the other hand, the non-linear entropy and the Gagliardo-Nirenberg inequality provide a Talagrand inequality which seems to be a new equivalence. Our method allows also to recover some results on the asymptotic behaviour of the associated gradient flows

A family of functional inequalities

For displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Lojasiewicz inequalities. In a second part, we specialise these inequalities to some classical geodesically convex functionals. For the Boltzmann entropy, we obtain the equivalence between logarithmic Sobolev and Talagrand's inequalities. On the other hand, the non-linear entropy and the Gagliardo-Nirenberg inequality provide a Talagrand inequality which seems to be a new equivalence. Our method allows also to recover some results on the asymptotic behaviour of the associated gradient flows

Definable Zero-Sum Stochastic Games

International audienceDefinable zero-sum stochastic games involve a finite number of states and action sets, reward and transition functions that are definable in an o-minimal structure. Prominent examples of such games are finite, semi-algebraic or globally subanalytic stochastic games. We prove that the Shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure. Definability in the same structure does not hold systematically: we provide a counterexample of a stochastic game with semi-algebraic data yielding a non semi-algebraic but globally subanalytic Shapley operator. %Showing the definability of the Shapley operator in full generality appears thus as a complex and challenging issue. } Our definability results on Shapley operators are used to prove that any separable definable game has a uniform value; in the case of polynomially bounded structures we also provide convergence rates. Using an approximation procedure, we actually establish that general zero-sum games with separable definable transition functions have a uniform value. These results highlight the key role played by the tame structure of transition functions. As particular cases of our main results, we obtain that stochastic games with polynomial transitions, definable games with finite actions on one side, definable games with perfect information or switching controls have a uniform value. Applications to nonlinear maps arising in risk sensitive control and Perron-Frobenius theory are also given

Researcher's Dilemma

We model academic competition as a game in which researchers ¯ght for priority. Researchers privately experience breakthroughs and decide how long to let their ideas mature before making them public, thereby establishing priority. In a two-researcher, symmetric environment, the resulting preemption game has a unique equilibrium. We study how the shape of the breakthrough distribution affects equilibrium maturation delays. Making researchers better at discovering new ideas or at developing them has contrasted effects on the quality of research outputs. Finally, when researchers have different innovative abilities, speed of discovery and maturation of ideas are positively correlated in equilibrium

Subgradient sampling for nonsmooth nonconvex minimization

Risk minimization for nonsmooth nonconvex problems naturally leads to firstorder sampling or, by an abuse of terminology, to stochastic subgradient descent. We establish the convergence of this method in the path-differentiable case, and describe more precise results under additional geometric assumptions. We recover and improve results from Ermoliev-Norkin by using a different approach: conservative calculus and the ODE method. In the definable case, we show that first-order subgradient sampling avoids artificial critical point with probability one and applies moreover to a large range of risk minimization problems in deep learning, based on the backpropagation oracle. As byproducts of our approach, we obtain several results on integration of independent interest, such as an interchange result for conservative derivatives and integrals, or the definability of set-valued parameterized integrals

Researcher's Dilemma

We model academic competition as a game in which researchers ¯ght for priority. Researchers privately experience breakthroughs and decide how long to let their ideas mature before making them public, thereby establishing priority. In a two-researcher, symmetric environment, the resulting preemption game has a unique equilibrium. We study how the shape of the breakthrough distribution affects equilibrium maturation delays. Making researchers better at discovering new ideas or at developing them has contrasted effects on the quality of research outputs. Finally, when researchers have different innovative abilities, speed of discovery and maturation of ideas are positively correlated in equilibrium

Researcher's Dilemma

We model academic competition as a game in which researchers ¯ght for priority. Researchers privately experience breakthroughs and decide how long to let their ideas mature before making them public, thereby establishing priority. In a two-researcher, symmetric environment, the resulting preemption game has a unique equilibrium. We study how the shape of the breakthrough distribution affects equilibrium maturation delays. Making researchers better at discovering new ideas or at developing them has contrasted effects on the quality of research outputs. Finally, when researchers have different innovative abilities, speed of discovery and maturation of ideas are positively correlated in equilibrium